Understanding Intersecting Linear Equations

We are given a pair of linear equations in two variables, and we need to determine the condition on the value of 'k' such that these lines intersect each other on a graph.

The two given linear equations are:

• Equation 1: \(kx + 3y + 1 = 0\)
• Equation 2: \(2x + y + 3 = 0\)

These equations are in the standard form \(ax + by + c = 0\).

Condition for Intersecting Lines

For a pair of linear equations given in the standard form:

• \(a_1x + b_1y + c_1 = 0\)
• \(a_2x + b_2y + c_2 = 0\)

The lines represented by these equations will intersect at exactly one point if and only if the ratio of the coefficients of \(x\) is not equal to the ratio of the coefficients of \(y\).

The algebraic condition for intersecting lines is:

\(\frac{a_1}{a_2} \ne \frac{b_1}{b_2}\)

Identifying Coefficients of the Given Equations

Let's identify the coefficients \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\) from our given equations:

• From Equation 1, \(kx + 3y + 1 = 0\), we have:
  • \(a_1 = k\)
  • \(b_1 = 3\)
  • \(c_1 = 1\)
• From Equation 2, \(2x + y + 3 = 0\), we have:
  • \(a_2 = 2\)
  • \(b_2 = 1\)
  • \(c_2 = 3\)

Applying the Intersection Condition for k

Now, we apply the condition for intersecting lines using the coefficients we just identified:

\(\frac{a_1}{a_2} \ne \frac{b_1}{b_2}\)

Substitute the values of \(a_1, a_2, b_1, b_2\) into the condition:

\(\frac{k}{2} \ne \frac{3}{1}\)

\(\frac{k}{2} \ne 3\)

To find the condition on \(k\), we need to isolate \(k\). We can do this by multiplying both sides of the inequality by 2:

\(k \ne 3 \times 2\)

\(k \ne 6\)

Conclusion on the Value of k

Thus, the pair of linear equations \(kx + 3y + 1 = 0\) and \(2x + y + 3 = 0\) will intersect each other if and only if the value of \(k\) is not equal to 6.

This means any value of \(k\) except 6 will result in these two lines intersecting at a unique point.

Revision Table: Conditions for Pairs of Linear Equations

Additional Information on Solutions of Linear Equations

When we talk about the solution to a pair of linear equations, we are referring to the point(s) \((x, y)\) that satisfy both equations simultaneously. Graphically, these are the points where the lines intersect.

• If the lines intersect at a single point, there is exactly one solution. This happens when \(\frac{a_1}{a_2} \ne \frac{b_1}{b_2}\). The system is called a consistent system.
• If the lines are parallel and distinct, they never intersect. There is no common point, so there is no solution. This happens when \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}\). The system is called an inconsistent system.
• If the lines are coincident (one lies exactly on top of the other), they intersect at every point. There are infinitely many common points, so there are infinitely many solutions. This happens when \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\). The system is called a consistent and dependent system.

Our problem specifically asked for the condition under which the lines intersect, which means we needed the condition for a unique solution, leading to \(k \ne 6\).